nLab
formal neighbourhood of the diagonal

Contents

Idea

Given any object XX in a cartesian monoidal category where formal completion is defined, then the formal neighbourhood of the diagonal of XX is the formal completion of the diagonal map Δ X:XX×X\Delta_X \colon X \longrightarrow X \times X.

Hence intuitively, the formal neighbourhood of the diagonal of XX is the space whose points are pairs of infinitesimally close points in XX.

Specifically for XX a scheme, then the formal neighbourhood of its diagonal is the formal scheme around Δ X\Delta_X. In this case the coequalizer of the two projections out of the formal neighbourhood is called the de Rham stack of XX. Moreover, the (sheaf of modules of) sections of the formal neighbourhood, with respect to one of the two projection maps, is the tangent complex of XX. This holds indeed more generally, at least also for XX a derived algebraic stack (Hennion 13).

More generally, see at infinitesimal disk bundle.

References

Last revised on January 9, 2016 at 17:38:39. See the history of this page for a list of all contributions to it.